{"id":"n-99e5bf113a0a46cd","revision":1,"first_index":0,"fingerprint":"f6-59d63bf33cbf8ad6","name":"Triangle read by rows, A007318 rows repeated three times.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["1"],["1"],["1","1"],["1","1"],["1","1"],["1","2","1"],["1","2","1"],["1","2","1"],["1","3","3","1"],["1","3","3","1"],["1","3","3","1"],["1","4","6","4","1"],["1","4","6","4","1"],["1","4","6","4","1"]],"degrees":[0,0,0,1,1,1,2,2,2,3,3,3,4,4,4],"row_sums":["1","1","1","2","2","2","4","4","4","8","8","8","16","16","16"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A152828","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159420,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-9670611e62c74d1a"]},{"id":159421,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-9670611e62c74d1a"]},{"id":159422,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-9670611e62c74d1a"]}],"region_exact":{},"incoming_edges":[{"id":"e-9670611e62c74d1a","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-99e5bf113a0a46cd","premise_count":0,"theorem":"polytool parent provers, route periodic_product: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"360b8fc223310478a8629fc551ea9a25eee15a616db5d801441552f53544d656","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[{"id":"e-131d94ab8eb56aaf","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-a5bdea4e5eb1caa8","conclusion_name":"shift_x_plus_one of Triangle read by rows, A007318 rows repeated three times."},{"id":"e-15d22f53a2fb1793","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-ae337235b402d9c7","conclusion_name":"basis_map_A013611 of Triangle read by rows, A007318 rows repeated three times."},{"id":"e-1fe5b61acc156f66","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-1cb9c12ec3656f95","conclusion_name":"basis_map_A013609 of Triangle read by rows, A007318 rows repeated three times."},{"id":"e-9d0df611e4535b39","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-7b34b2b914f7d002","conclusion_name":"basis_map_A013610 of Triangle read by rows, A007318 rows repeated three times."},{"id":"e-be32ce4d8d6a286f","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-9eca18e169011a2f","conclusion_name":"basis_map_A008277 of Triangle read by rows, A007318 rows repeated three times."},{"id":"e-345c88cc98fbdbfc","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-a5bdea4e5eb1caa8","conclusion_name":"shift_x_plus_one of Triangle read by rows, A007318 rows repeated three times."},{"id":"e-809cfb1679ce14dc","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-39d43b83afedcf47","conclusion_name":"one_plus_derivative of Triangle read by rows, A007318 rows repeated three times."},{"id":"e-916a7875f7a82d69","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-53b76fba1338a47f","conclusion_name":"derivative of Triangle read by rows, A007318 rows repeated three times."}],"used_by_total":8,"links":[{"id":"l-b7e4710714f3c0be","kind":"fingerprint_related","node_a_id":"n-99e5bf113a0a46cd","node_b_id":"n-1cb9c12ec3656f95","fingerprint":"f6-59d63bf33cbf8ad6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-59d63bf33cbf8ad6"},"other_id":"n-1cb9c12ec3656f95","other":{"id":"n-1cb9c12ec3656f95","name":"basis_map_A013609 of Triangle read by rows, A007318 rows repeated three times.","first_index":0,"fingerprint":"f6-59d63bf33cbf8ad6","polytool_family":null,"degree_pattern":"irregular","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["1"]],"card_rows":[["1"],["1"],["1"],["2","2"]],"row_sums_preview":["1","1","1","4","4","4","16","16"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-99e5bf113a0a46cd","steps":["basis A013609"],"edges":["e-1fe5b61acc156f66"],"text":"A152828 \u2192 basis A013609"},"headline":"A152828 \u2192 basis A013609","chain":null}},{"id":"l-db72ad4283a52f02","kind":"fingerprint_related","node_a_id":"n-99e5bf113a0a46cd","node_b_id":"n-53b76fba1338a47f","fingerprint":"f6-59d63bf33cbf8ad6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-59d63bf33cbf8ad6"},"other_id":"n-53b76fba1338a47f","other":{"id":"n-53b76fba1338a47f","name":"derivative of Triangle read by rows, A007318 rows repeated three times.","first_index":0,"fingerprint":"f6-59d63bf33cbf8ad6","polytool_family":null,"degree_pattern":"irregular","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],[]],"card_rows":[["1"],["1"],["1"],["2","2"]],"row_sums_preview":["0","0","0","1","1","1","4","4"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-99e5bf113a0a46cd","steps":["D"],"edges":["e-916a7875f7a82d69"],"text":"A152828 \u2192 D"},"headline":"A152828 \u2192 D","chain":null}},{"id":"l-22cdb902b30da04b","kind":"oeis_match","node_a_id":"n-99e5bf113a0a46cd","node_b_id":null,"fingerprint":null,"oeis_id":"A152828","in_catalog":1,"target":"reversed_triangle","row_alignment":"node_lengths","rows_matched":15,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":15,"target":"reversed_triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A152828","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-5b251d9fdeff06b1","kind":"oeis_match","node_a_id":"n-99e5bf113a0a46cd","node_b_id":null,"fingerprint":null,"oeis_id":"A173862","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A173862","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-7607975da552daf2","kind":"oeis_match","node_a_id":"n-99e5bf113a0a46cd","node_b_id":null,"fingerprint":null,"oeis_id":"A152828","in_catalog":1,"target":"triangle","row_alignment":"node_lengths","rows_matched":15,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":15,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A152828","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A152828","relation":"identification"},{"oeis_id":"A152828","relation":"match_reversed_triangle"},{"oeis_id":"A173862","relation":"match_row_sums"},{"oeis_id":"A152828","relation":"match_triangle"}],"bfiles":[{"node_id":"n-99e5bf113a0a46cd","oeis_id":"A152828","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":76,"terms_matched":45,"oeis_terms_agreeing":95,"oeis_terms_available":95,"term_count":1001,"raw_size":8282},{"node_id":"n-99e5bf113a0a46cd","oeis_id":"A152828","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":76,"terms_matched":45,"oeis_terms_agreeing":95,"oeis_terms_available":95,"term_count":1001,"raw_size":8282},{"node_id":"n-99e5bf113a0a46cd","oeis_id":"A173862","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":53,"oeis_terms_available":53,"term_count":200,"raw_size":2972}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1;\n  2: 1;\n  3: 1, 1;\n  4: 1, 1;\n  5: 1, 1;\n  6: 1, 2, 1;\n  7: 1, 2, 1;\n  8: 1, 2, 1;\n  9: 1, 3, 3, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 3, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (1 + x)*v[i - 3]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A152828","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1;\n  2: 1;\n  3: 1, 1;\n  4: 1, 1;\n  5: 1, 1;\n  6: 1, 2, 1;\n  7: 1, 2, 1;\n  8: 1, 2, 1;\n  9: 1, 3, 3, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 3, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (1 + x)*v[i - 3]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A152828","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1;\n  2: 1;\n  3: 1, 1;\n  4: 1, 1;\n  5: 1, 1;\n  6: 1, 2, 1;\n  7: 1, 2, 1;\n  8: 1, 2, 1;\n  9: 1, 3, 3, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 3, x]];\nTable[Reverse[CoefficientList[P[n, x], x]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (1 + x)*v[i - 3]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-99e5bf113a0a46cd","steps":[],"edges":[],"text":"A152828"},"headline":"A152828: Triangle read by rows, A007318 rows repeated three times.","chain":null}
